A336083 Decimal expansion of the arclength on the unit circle such that the corresponding chord separates the interior into segments having 3 = ratio of segment areas; see Comments.

2, 3, 0, 9, 8, 8, 1, 4, 6, 0, 0, 1, 0, 0, 5, 7, 2, 6, 0, 8, 8, 6, 6, 3, 3, 7, 7, 9, 3, 1, 3, 6, 2, 4, 8, 4, 6, 1, 1, 1, 9, 9, 6, 4, 5, 8, 5, 8, 8, 3, 1, 0, 3, 7, 5, 4, 5, 3, 1, 5, 2, 9, 3, 1, 9, 2, 7, 1, 9, 2, 8, 5, 8, 0, 2, 6, 6, 5, 2, 0, 9, 3, 9, 1, 3, 3

Offset

1,1

Comments

Suppose that s in (0,Pi) is the length of an arc of the unit circle. The associated chord separates the interior into two segments. Let A1 be the area of the larger and A2 the area of the smaller. The term "ratio of segment areas" means A1/A2. See A336073 for a guide to related sequences.

Equals the median of the probability distribution function of angles of random rotations in 3D space uniformly distributed with respect to the Haar measure, i.e., the solution x to Integral_{t=0..x} ((1 - cos(t))/Pi) dt = 1/2 (see Reynolds, 2017; cf. A086118, A361605). - Amiram Eldar, Mar 17 2023

Links

Table of n, a(n) for n=1..86.

Marc B. Reynolds, Volume element of SO(3) and average uniform random rotation angle, 2017.

Formula

Equals d+Pi/2 = A003957 + A019669, where d is the Dottie number. - Gleb Koloskov, Feb 21 2021

Example

arclength = 2.3098814600100572608866337793136248461119964...

Mathematica

k = 3; s = s /. FindRoot[(2 Pi - s + Sin[s])/(s - Sin[s]) == k, {s, 2}, WorkingPrecision -> 200]
RealDigits[s][[1]]

Prog

(PARI) d=solve(x=0, 1, cos(x)-x); d+Pi/2 \\ Gleb Koloskov, Feb 21 2021

Crossrefs

Cf. A336073, A003957, A019669.

Cf. A086118, A361605.

Sequence in context: A098989 A175315 A180186 * A256294 A279412 A012399

Adjacent sequences: A336080 A336081 A336082 * A336084 A336085 A336086

Keyword

nonn,cons

Author

Clark Kimberling, Jul 11 2020

Status

approved